This paper presents a new framework that aims to improve the efficiency of time response analysis for nonlinear dynamical systems by combining conventional time integration methods with Proper Generalized Decomposition (PGD). The PGD approach utilizes low-dimensional subspaces of the time response to approximate the solution as a low-order separated representation of spatial and temporal components, with the Galerkin projection employed to formulate subproblems for each component. The subproblem for spatial basis is viewed as computing a reduced-order criterion, and the temporal problem projected to a subspace spanning this criterion uses time integration to obtain time coefficients. During the time integration, the spatial modes obtained from the calculation of the previous step are used as a reduced basis, and additional spatial modes are added until the residual of equations of motion satisfy the target tolerance. Numerical examples demonstrate that the proposed method allows significant computational savings compared to conventional time integration methods while accurately reflecting the nonlinear behavior.
Optical imaging-based remote sensing of the mechanical motion of an object is presented utilizing phase-based motion magnification (PMM) technology that amplifies subtle movement of the object invisible to the naked eye. In case that the oscillatory motion of an object is very small compared to the pixel size of optical image, the resulting motion measurement is inaccurate due to the low signal-to-noise ratio (SNR). To overcome this limitation, various vibration measurement method using phase information of the optical image is known to be more robust to noise and lighting conditions as well, however the existing PMM technology has several inherent issues to be addressed such as phase ambiguity. In this context, this paper focuses on the improvement of motion measurement accuracy by overcoming the phase ambiguity, in turn, limitation of magnification range of the remote motion sensing in existing PMM technique. The proposed PMM utilizes Gabor-like complex steerable pyramid (CSP) to expand the magnification range while retaining the local characteristics of CSP. Specifically, phase unwrapping is employed to resolve the discontinuity of the phase, and the novel envelope shift method is newly applied on the top of phase unwrapping to extend the magnification range. We verified the proposed method through the experiment of the vibrating structure. The proposed method reduces the ringing artifact and blurring of the magnified motion of the target structure. The accuracy of the proposed method was compared to the physical accelerometer measurements and previous methods. The experimental verification showed that the proposed method has accuracy of more than 90% with reduced motion artifact, while the original PMM has an accuracy of about 67% to 87% depending on the magnification factor, which demonstrate accuracy and robustness of the proposed method.
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